Chapter 13

. . . And Chance

I returned and saw under the sun, that the race is not to the
swift, nor the battle to the strong, neither bread to the wise, nor
yet riches to men of understanding, nor yet favour to men of skill;
but time and chance happeneth to them all.

-- Ecclesiastes 9.11

PART 1 - SUNK COSTS

So far, I have introduced time into the economy,
but not uncertainty; everything always comes out as expected. The
real world is not so simple. One of the consequences of uncertainty
is the possibility of mistakes; another is the problem of what to do
about them.

You see an advertisement for a shirt sale at a
store 20 miles from your home. You were planning to buy some new
shirts, and the prices are substantially lower than in your local
clothing store; you decide the savings are enough to make it worth
the trip. When you arrive, you discover that none of the shirts on
sale are your size; the shirts that are your size cost only slightly
less than in your local store. What should you do?

You should buy the shirts. The cost of driving to
the store is a sunk cost--once incurred, it cannot be recovered. If
you had known the prices before you left home, you would have
concluded that it was not worth making the trip--but now that you
have made it, you must pay for it whether or not you buy the shirts.
Sunk costs are sunk costs.

There are two opposite mistakes one may make with
regard to sunk costs. The first is to treat them as if they were not
sunk--to refuse to buy the shirts because their price is not low
enough to justify the trip even though the trip has already been
made. The second is to buy the shirts even when they are more
expensive than in your local store, on the theory that you might as
well get something for your trip. The something you are getting in
this case is less than nothing. This is known as throwing good money
after bad.

When, as a very small child, I quarrelled with my
sister and then locked myself in my room, my father would come to the
door and say, "Making a mistake and not admitting it is only hurting
yourself twice." When I got a little older, he changed it to "Sunk
costs are sunk costs."

In discussing firms' cost curves, one should
distinguish between fixed costs and sunk costs--while the same costs
are often both fixed and sunk, they need not always be. Fixed costs
are costs you must pay in order to produce anything--the limit of
total cost as a function of quantity when quantity approaches zero.
One could imagine a case where such costs were fixed but not sunk,
either because the necessary equipment could be resold at its
purchase price or because the equipment was rented and the rental
could be terminated any time the firm decided to stop
producing.

The significance of sunk costs is that a firm will
continue to produce even when revenue does not cover total cost,
provided that it does cover nonsunk costs (called recoverable costs),
since nonsunk costs are all the firm can save by closing down. All
costs, ultimately, are opportunity costs--the cost of doing one thing
is not being able to do something else. Once a factory is built, the
cost of continuing to run it does not include what was spent building
it, since whatever you do you will not get that back. It does include
the cost of not selling it to someone else--which may be more or less
than the cost of building it, depending on whether the value of such
factories has gone up or down since it was built.

In deriving the supply curve for a competitive
industry with open entry in Chapter 9, we saw that firms would always
produce at the minimum of average cost, where it crossed marginal
cost. The reason was that if, at the quantity for which marginal cost
equaled price (where profit is maximized for a price taker), price
were above average cost, economic profit would be positive; it would
pay other firms to enter the industry. They would do so until price
was driven down to the point where it equaled both MC and AC, which
occurs where they cross at the minimum of AC.

Does the relevant average cost include sunk costs?
That depends on whether we are approaching the equilibrium from above
or below and on how long a time we consider. If prices start out
above the equilibrium price, firms will only enter the industry as
long as the price is above average cost including sunk cost--costs
are not sunk until they are incurred, and the new firm starts out
with the option of not incurring them. The equilibrium will be
reached when price equals average total cost.

If we approach the equilibrium from below--if
there are too many firms (perhaps because demand has recently fallen)
and price is insufficient to cover even the average of recoverable
costs--firms will leave the market. They will continue to do so until
price gets up to average recoverable cost.

If the assets bought with the sunk costs
(factories, say) wear out over time, then the number of factories
will gradually decline and the price will gradually rise. Until it
reaches average total cost, nobody will build any new factories.
Eventually price will be equal to average total cost, just as it was
when we reached the equilibrium from above, but it may take much
longer to get there; it usually takes longer to wear out a factory
than to build one.

In the next two sections, I will work through the
logic of such situations in some detail while trying to show how it
is related to the logic of a different sort of situation that was
briefly discussed several chapters ago.

Upside, Downside, Cost Equals Price

In analyzing the industry supply curve in Chapter
9, I assumed an unlimited number of potential firms, all with the
same cost curve; if existing firms make a profit, new firms come into
existence until the profit is competed down to zero.

One objection to this that I discussed is that
firms are not all identical. Some power companies own special pieces
of real estate--Niagara Falls, for example--not available to their
competitors. Some corporations are run by superb managers or possess
the services of an inventive genius such as Browning or Kloss. Surely
such fortunate firms can, as a result, produce their output at a
lower cost than others--and can therefore make profits at a price at
which it does not pay less fortunate firms to enter the
industry.

But although firms that have, in this sense, low
cost curves appear to make positive profits when less fortunate firms
just barely cover their costs, that is an illusion. One should
include in cost the cost of using the special assets (location,
administrator, inventor, or whatever) that give that firm its
advantage. The value of those assets is what the firm could sell them
for or, in the case of human assets, what a competitor would pay to
hire them away. One of the firm's (opportunity) costs of operating is
not selling out, and one of the costs to an inventor of running his
own firm is not working for someone else. If the possession of those
special assets gives the firm an additional net revenue of, say,
$100,000/year (forever--or almost), then the market value of those
assets is the present value of that income stream. The interest on
that present value is then the same $100,000/year. Since giving up
that interest is one of the costs to the firm of staying in business,
the firm should subtract it from revenue in calculating its economic
profit.

Suppose, for example, that the firm is making an
extra $100,000/year as a result of owning its special asset and that
the interest rate is 10 percent. The present value of a permanent
income stream of $100,000/year is $1,000,000, and the interest on
$1,000,000 is $100,000. By using the asset this year, the firm gives
up the opportunity to sell it and collect interest on the money it
would get for it. We should include $100,000/year as an additional
cost--forgone interest. Doing so reduces the profit of the firm to
zero--the same as the profit of an ordinary firm. In one sense, this
argument is circular; in another sense, it is not.

The same argument applies in the opposite
direction to firms whose revenues fail to cover their sunk costs
(firms whose revenues fail to cover their recoverable costs go out of
business). Suppose a widget factory costs $1,000,000 to build and
lasts forever; further suppose the interest rate is 10 percent, so
that the factory must generate net revenue of $100,000/year to be
worth building. At the time the factory is built, the price of
widgets is $1.10/widget. The factory can produce 100,000 widgets per
year at a cost (not including the cost of building the factory) of
$0.10/widget, so it is making $100,000/year--just enough to justify
the cost of building it. Further suppose that the factory can be used
for nothing but building widgets; its scrap value is zero.

The invention of the fimbriated gidget drastically
reduces the demand for widgets. Widget prices fall from $1.10 to
$0.20. At a price of $0.20, the firm is netting only $10,000/year on
its $1,000,000 investment. So are all the other (identical) firms.
Are they covering costs?

The factory is a sunk cost from the standpoint of
the industry, but any individual firm can receive its value by
selling it to another firm. How much will it sell for? Since it
generates an income of $10,000/year and since at an interest rate of
10 percent an investment of $100,000 can generate the same income,
the factory will sell for $100,000. So the cost of not selling it is
$100,000--and the annual cost of not selling it is $10,000, the
interest forgone. Ten thousand dollars is the firm's revenue net of
costs before subtracting the cost of the factory, so net revenue
after subtracting the cost of the factory--economic profit--is
zero.

Again the argument is circular but not empty,
since it tells us, among other things, what determines the price of a
factory in a declining industry. In the case I have just described,
the firm loses $900,000 the day the price of widgets drops, since
that is the decrease in the value of its factory. Thereafter it just
covers costs, as usual.

The assumptions used in this example, although
useful for illustrating the particular argument, are not quite
consistent with rational behavior. In the market equilibrium before
the price drop, economic profit was zero. That is an appropriate
assumption for the certain world of Chapters 1-12, but not for the
uncertain world we are now discussing. If there is some possibility
of prices falling, then firms will bear sunk costs only if the
average return justifies the investment. Prices must be high enough
that the profit if they do not fall balances the loss if they do. The
zero-profit condition continues to apply, but only in an average
sense--if the firms are lucky, they make money; if they are unlucky,
they lose it. On average they break even. This point will be
discussed at greater length later in the chapter.

Sunk Costs in the Shipping Industry

You may find it helpful to work through another
example. Consider ships. Suppose that the total cost of building a
ship is $10,000,000. For simplicity we assume that operating costs
and the interest rate are both zero. Each ship lasts twenty years and
can transport 10,000 tons of cargo each year from port A to port B.
We assume, again for simplicity, that the ships all come back from B
to A empty. It takes a year to build a ship. The demand curve for
shipping cargo is shown in Figure 13-1a.

We start with our usual competitive
equilibrium--price equals average cost. There are 100 ships and the
cost for shipping cargo is $50/ton. Each ship makes $500,000 a year;
at the end of twenty years, when the ship collapses into a pile of
rust, it has just paid for itself. Every year five ships are built to
replace the five that have worn out. If the price for shipping were
any higher, it would pay to build more ships, since an investment of
$10,000,000 would produce a return of more than $10,000,000; if it
were lower, no ships would be built. The situation is shown in Figure
13-1a.

Figure 13-1b shows the effect of a sudden increase
in the demand for shipping--from D to D'. In the short run, the
supply of shipping is perfectly inelastic, since it takes a year to
build a ship. The price shoots up to P1, where the new demand curve
intersects the short-run supply curve.

Supply and demand curves for shipping, showing the
effect of an unanticipated increase in demand. Figure 13-la shows the
situation before the increase and Figure 13-lb after. The horizontal
axis shows both quantity of cargo carried each year and the
equivalent number of ships. The short-run supply curve is vertical at
the current number of ships (and amount of cargo they carry). The
long-run supply curve is horizontal at the cost of producing shipping
(the annualized cost of building a ship divided by the number of tons
it carries).

Shipyards immediately start building new ships. At
the end of a year, the new ships are finished and the price drops
back down to the old level. Figure 13-2 shows the sequence of events
in the form of a graph of price against time.

Looking again at Figure 13-lb, note that it has
two supply curves--a vertical short-run supply curve and a horizontal
long-run supply curve. No ships can be built in less than a year, so
there is no way a high price can increase the supply of shipping in
the short run. Since operating costs are, by assumption, zero, it
pays shipowners to operate the ships however low the price; there is
no way a low price can reduce the supply of shipping in the short
run. So in the short run, quantity supplied is independent of price
for any price between zero and infinity.

The situation in the long run is quite different.
At any price where ships more than cover their construction cost, it
pays to build ships; so in the long run, the industry will produce an
unlimited quantity of shipping at any price above P0 = $50/ton. As
ships are built, the short-run supply curve shifts out. At any price
below P0, building a ship costs more than the ship is worth, so
quantity supplied falls as the existing ships wear out. So the
long-run supply curve is horizontal. It is worth noting that on the
"up" side--building ships--the long run is a good deal shorter than
on the "down" side.

Suppose that instead of the increase in demand
shown in Figure 13-1b, there is instead a decrease in demand, from D
to D", as shown in Figure 13-3a. Price drops. Since there are no
operating costs, existing ships continue to carry cargo as long as
they get any price above zero. The price is at the point where the
old (short-run) vertical supply curve intersects the new demand curve
(A).

Building a ship is now unprofitable, since it will
not, at the new price, repay its construction costs. No ships are
built. Over the next five years, 25 ships wear out, bringing the
long-run quantity supplied (and the short-run supply curve) down to a
point where the price is again $50/ton (B). Figure 13-3b shows how
the price and the number of ships change with time.

There is one thing wrong with this story. The
initial equilibrium assumed that the price of shipping was going to
stay the same over the lifetime of a ship--that was why ships were
produced if and only if the return at current prices, multiplied by
the lifetime of the ship, totaled at least the cost of production.
The later developments assumed that the demand curve, and hence the
price, could vary unpredictably.

The effect of an unexpected decrease in demand for
shipping. Figure 13-3a shows the situation after the demand curve
shifts. Figure 13-3b shows the resulting pattern of prices over
time.

A possible pattern of freight rates over time.
Unlike Figure 13-2, this figure assumes that the producers expect
unpredictable shifts in demand. The average return from carrying
freight must be enough to just cover the costs.

If shipowners expect random changes in future
demand and believe that future decreases will be at least as frequent
and as large as future increases, the price at which they are just
willing to build will be more than $50/ton. Why? Because ships can be
built quickly, so that the gain from an increase in demand is
short-lived, but wear out slowly, so that the loss from a decrease in
demand continues for a long time. Compare the short period of high
prices in Figure 13-2 with the long period of low prices in Figure
13-3b. If the current price is high enough (Pe on Figure 13-4) that
any increase causes ships to be built, then an increase in demand
will hold prices above Pe for only a year. A decrease can keep prices
below Pe for up to twenty years. If Pe were equal to $50/ton, the
price at which ships exactly repay the cost of building them, the
average price would be lower than that and ships, on average, would
fail to recover their costs. So Pe must be above $50/ton.

This is the same point that I made earlier in
describing the effect of sunk costs in the widget industry. In order
to make the behavior of the shipowners rational, we must assume that
they do not start building ships until the price is high enough that
the profits if demand does not fall make up for the losses if it
does. The pattern of price over time in the industry then looks
something like Figure 13-4.

How to Lie While Telling the
Truth---

A True Story

Many years ago, while spending a summer in
Washington, I came across an interesting piece of economic analysis
involving these principles. The congressman I was working for had
introduced a bill that would have abolished a large part of the farm
program, including price supports for feed grains (crops used to feed
animals). Shortly thereafter the agriculture department released a
"study" of the effects of abolishing those particular parts of the
farm program. Their conclusion, as I remember, was that farm income
would fall by $5 billion while the government would save only $3
billion in reduced expenditure, for a net loss of $2 billion.

The agriculture department's calculations
completely ignored the effect of the proposed changes on
consumers--although the whole point of the price support program was
(and is) to raise the price of farm products and thus of food. Using
the agriculture department's figures, the proposed abolition would
have saved consumers (as I remember) about $7 billion, producing a
net gain of $5 billion. The agriculture department, which of course
opposed the proposed changes, failed to mention that implication of
its analysis.

Another part of the report asserted that the
abolition of price supports on feed grains would drive down the
prices of the animals that consumed them. It went on to say that the
price drop would first hit poultry producers, then producers of pork
and lamb, and finally beef producers. All of this, to the best of my
knowledge, is correct. The conclusion that appears to follow is that
poultry producers will be injured a great deal by the abolition, lamb
and pork producers somewhat less, and beef producers injured least of
all. This is almost the precise opposite of the truth.

If you think about the situation for a moment, you
should be able to see what is happening. Removing price supports on
feed grains lowers the cost of production for poultry, pork, lamb,
and beef--feed grains are probably the largest input for producing
those foods. In the case of poultry, the flocks can be rapidly
increased, so the poultry producers will receive an above-normal
profit (cost of production has fallen, price of poultry has not) for
only a short time. Once the flocks have increased, the price of
chickens falls and the return to their producers goes back to normal.
The herds of pigs and sheep take longer to increase, so their
producers get above-normal returns for a longer period, and the beef
producers get them for longer still. The situation is just like the
situation of the shipowners when demand increases, except that there
is a drop in production cost rather than an increase in the demand
schedule. The agriculture department appeared to be saying that the
beef producers would receive the least injury and the poultry
producers the greatest injury from the proposed change; what their
analysis actually implied was that the beef producers would receive
the largest benefit and the poultry producers the smallest
benefit.

PART 2 - LONG-RUN AND

SHORT-RUN COSTS

So far, we have been analyzing the influence of
uncertainty on prices by taking account of the effect of sunk costs
on the behavior of profit-maximizing firms. A more technical
description of what we are doing is that we are analyzing the effect
of uncertainty in terms of Marshallian quasi-rents-- "Marshallian"
because this approach, along with much of the rest of modern
economics, was invented by Alfred Marshall about a hundred years ago
and "quasi-rents" because the return on sunk costs is in many ways
similar to the rent on land. Both can be viewed as the result of a
demand curve intersecting a perfectly inelastic supply
curve--although in the case of sunk costs, the supply curve is
inelastic only in the short run.

The more conventional way of analyzing these
questions is in terms of short-run and long-run cost curves and the
resulting short-run and long-run supply curves. I did not use that
approach in Chapter 9, where supply curves were deduced from cost
curves, and so far I have not used it here. Why?

The reason for ignoring the distinction between
long-run and short-run costs in Chapter 9 was explained there; in the
unchanging world we were analyzing, long run and short run are the
same. The reason I did not introduce the ideas of this chapter in
that form is that the way in which I did introduce it provides a more
general and more powerful way of analyzing the same questions. It is
more general because it allows us to consider productive assets--such
as ships and factories--with a variety of lifetimes and construction
times, not merely the extreme (and arbitrary) classes of "short-" and
"long-" lived. It is more powerful because it not only gives us the
long-run and short-run supply curves but also shows what happens in
between, both to the price of the productive assets and to the price
of the goods they produce.

The simplest way to demonstrate all of this--and
to prepare you for later courses that will assume you are familiar
with the conventional approach--is to work out the short-run/long-run
analysis as a special case of the approach we have been following.
While doing so, we will also be able to examine some complications
that have so far remained hidden behind the simplifying assumptions
of our examples.

Factory Size and the Short-Run Supply
Curve

We start with an industry. Since we used up our
quota of widgets earlier in the chapter, we will make it the batten
industry. Battens are produced in batten factories. There are many
batten firms, so each is a price taker. A firm entering the
industry--or a firm already in the industry that is replacing a
worn-out factory--must choose what size factory to build. A small
factory is inexpensive to build but expensive to operate--especially
if you want to produce a large amount of output. Larger factories
cost more to build but are more efficient for producing large
quantities. A firm can only operate one factory at a time.

Figures 13-5 through 13-7 show the cost curves for
three different factories. The first costs $1 million to build, the
second $3 million, and the third $5 million. A factory has no scrap
value, so the investment is a sunk cost. Each factory lasts ten
years. The interest rate is zero, so the annual cost associated with
each factory is one tenth the cost of building it. One could easily
enough do the problem for a more realistic interest rate, but that
would complicate the calculations without adding anything
important.

Total cost is the sum of fixed cost and variable
cost. The figures are drawn on the assumption that the only fixed
cost in producing battens is the cost of building the factory; all
other costs are variable. Since this implies that the fixed cost and
the sunk cost are identical, so are variable cost (total cost minus
fixed cost) and recoverable cost (total cost minus sunk cost). The
figures show average variable cost (AVC); it might just as well have
been labeled ARC for "average recoverable cost."

Each pair of figures shows four cost curves--total
cost (TC), marginal cost (MC), average cost (AC), and average
variable cost (AVC). Total cost includes the (annualized) cost of the
factory; since that is assumed to be the only fixed cost, total cost
at a quantity of zero is the annualized cost of the
factory--$100,000/year on Figure 13-5b. Since average cost is defined
as total cost over quantity, it too includes the cost of the factory.
Average variable cost, on the other hand, does not include the cost
of the factory, since that is fixed.

So far as marginal cost is concerned, it does not
matter whether or not we include the cost of the factory. Marginal
cost is the slope of total cost; adding a constant term to a function
simply shifts it up without affecting its slope.

Suppose the batten firm has built the factory of
Figure 13-5. The market price of a batten is P; the firm must decide
how many to produce each year. Just as in Chapter 9, the firm
maximizes its profit by producing the quantity for which MC = P,
provided that at that quantity it is not losing money.

In Chapter 9, we could see whether the firm was
making or losing money by comparing price to average cost; if average
cost is greater than price, then profit is negative and the firm
should go out of business. This time we have two average costs--AC
and AVC. Which should we use?

We should use AVC. The firm already has the
factory; it is deciding whether or not to shut it down. If the firm
shuts down the factory, it will not get back the money that was spent
to build it--that is a sunk cost. What it will save is its variable
cost. If the savings from shutting down the factory are greater than
the loss from no longer having any battens to sell, then the factory
should be shut down. Otherwise it should continue to operate. So as
long as price is greater than average variable cost, the firm
continues to operate the factory, producing the quantity for which
marginal cost equals price. If price is lower than average cost, the
factory is not paying back its cost of construction and should never
have been built--but it is too late to do anything about that. Sunk
costs are sunk costs.

The curves labeled S1-S3 on Figures 13-5 through 13-7
are the supply curves implied by the previous two paragraphs. Each S
runs along the marginal cost curve, starting at its intersection with
average variable cost. For any price lower than that, quantity
supplied is zero.

The Long-Run Supply Curve

These are the short-run supply curves. They
correctly describe the behavior of a firm that already owns a
functioning factory. But in the long run, factories wear out and must
be replaced. A firm that is about to build a factory is in a
different situation, in two respects, from a firm that already has a
factory. First, the cost of building the factory is not yet sunk--the
firm has the alternative of not building and not producing. The firm
will build only if it expects price to be above average
cost--including in the average the cost of building the
factory.

The second difference is that a firm about to
build can choose which size of factory it prefers. Its choice will
depend on what the price is. So the long-run supply curve must take
account of the relation between the price of battens and the size of
the factories in which they will be produced.

How do we find the long-run supply curve of a
firm? We consider a firm that is about to build a factory and expects
the market price of battens to remain at its present level for at
least the next ten years--the lifetime of the factory. The firm's
long-run supply curve is then the relation between the quantity the
firm chooses to produce and the price.

We solve the problem in two steps. First we figure
out, for each size of factory, how many battens the firm will produce
if it decides to build a factory of that size. Then we compare the
resulting profits, in order to find out which factory the firm will
choose to build. Once we know which factory the firm chooses to build
and how much a firm with a factory of that size chooses to produce,
we know quantity supplied at that price. Repeat the calculation for
all other prices and we have the firm's long-run supply curve.

Figures 13-5 through 13-7 show the calculations
for a price of $10/batten. As we already know, if a price-taking firm
produces at all, it maximizes its profit by producing a quantity for
which MC = P. So for each size of factory, a firm that chose to build
that factory would produce the quantity for which marginal cost was
equal to price.

Having done so, what would the firm's profit be?
Profit per unit is simply price minus average cost. The firm should
include the cost of building the factory in deciding which factory to
build, so the relevant average is average cost, not average variable
cost. Total profit is profit per unit times number of units--the
shaded rectangle in each figure. It is largest for Figure 13-6, so
the firm builds a $3 million factory and produces that quantity for
which, in such a factory, price equals marginal cost.

Figure 13-8 shows the result of repeating the
calculations for many different prices. As I have drawn the curves,
the less expensive factories have a lower average cost for low levels
of output and a higher average cost for high levels. The result is
that as price (and quantity) increase, so does the optimal size of
the factory. The long-run supply curve for the firm (Figure 13-8b) is
then pieced together from portions of the short-run supply curves of
Figures 13-5 through 13-7. In doing so, we limit ourselves to the
part of each short-run supply curve above the corresponding average
cost (AC not AVC), since that is the long-run supply curve for that
size of factory. We end up with the long-run supply curve for a firm
that is free to vary factory size as well as other inputs.

Looking at Figure 13-8b, we see that the smallest
size of factory is irrelevant to the firm's supply curve, since there
is no price of battens at which it would be worth building such a
factory. If the market price is below P0, none of the three sizes of
factory can make enough money to justify the cost of building it, so
the firm produces nothing. For prices between P0 and
P1 on Figure
13-8b, the firm maximizes its profit by building a $3 million factory
and producing the quantity for which the marginal cost (MC2 on Figure
13-6a) equals the price. For prices above P1, it does better building a
$5 million factory and producing along the MC3 curve of Figure 13-7a.
So S is the firm's long-run supply curve.

An alternative way of deriving the long-run supply
curve of the firm is to consider the factory itself as one more input
in the production function. Just as in Chapter 9, one then calculates
the lowest cost bundle of inputs for each level of output; the result
tells you, for any quantity of output, how much it costs to produce
and what inputs--including what size of factory--you should use. You
then go on to calculate average cost (the same curve shown on Figure
13-8a), marginal cost, and the supply curve. Since we are considering
the long-run supply curve, we are (temporarily) back in the
unchanging world of Chapters 1-11.

Figure 13-9a shows what the firm's long-run
average cost curve would be like if, instead of limiting the firm to
only three sizes of factory, we allowed it to choose from a
continuous range of factory sizes. The solid line LAC on the figure
is the resulting long-run average cost curve; the gray lines are
average cost curves for several different factory sizes, including
those shown on Figures 13-5 through 13-7. Since for any quantity, the
firm chooses that factory size which produces that quantity at the
lowest possible cost, the average cost curve for a factory can never
lie below the average cost curve for the firm. Every point on the
firm's long-run average cost curve is also on the average cost curve
for some size of factory--the size the firm chooses to build if it
expects to produce that quantity of output. The result is what you
see on Figure 13-9a; the average cost curves for the different
factory sizes lie above the firm's long-run average cost curve and
are tangent to it.

One feature of figures such as 13-9a that some
people find puzzling is that the point where a factory average cost
curve touches the firm's long-run average cost curve is generally not
at the minimum average cost for that size of factory. AC1, for
example, touches LAC not at point B, which is its minimum, but at
point A, and similarly for all the others except AC2. Mathematically,
the reason for this is quite simple. AC1 is tangent to LAC at point
A. At the point of tangency, the two curves have the same slope.
Unless LAC is at its minimum--as it is at point C, where it touches
AC2--its slope is not zero. Since the slope of LAC is not zero at the
point of tangency, neither is the slope of AC1; so AC1 cannot be at
its minimum. The same applies to all of the points of tangency except
C.

As I have commented before, one can read through a
proof without ever understanding why the conclusion is correct; for
some of you, the previous paragraph may be an example of that.
Another way of putting the argument is to point out that while the
firm that chooses to produce quantity QA could lower its average cost
by expanding output to QB, it would then be producing a larger
quantity; if it wished to produce that quantity, it could do so at an
even lower average cost by using a bigger factory. B shows the
minimum average cost for producing in a $1 million factory. It does
not show the minimum average cost for producing a quantity QB, so it
does not show what the average cost would be for a firm that wished
to produce that quantity and was free to build whatever size factory
it preferred. Similarly, D is the minimum point on AC3, but there is
another (unlabeled) average cost curve lying below it, providing a
lower cost way of producing QD--at point F.

Figure 13-9a shows the short-run and long-run
average cost curves for a firm that can choose from a continuous
range of factory sizes. Figure 13-9b shows the long-run average cost
curve and the long-run supply curve for such a firm. Every time the
price goes up a little, the optimal size of factory shifts up as
well. The result is the smooth supply curve of Figure 13-9b.

In Chapter 9, after finding the supply curve for a
firm, we went on to find the supply curve for an industry made up of
many such firms. We can do the same thing here. In the short run, the
number of factories is fixed; there is not enough time to build more
or for existing factories to wear out. So the short-run supply curve
for the industry is simply the horizontal sum of the short-run supply
curves for all the existing factories--just as in the case of the
competitive industry with closed entry discussed in Chapter 9.

In the long run, the number of factories can vary;
firms may build new factories or fail to replace existing factories
as they wear out. Unless there are barriers to entry, such as laws
against building new factories, we are in the second case of Chapter
9--a competitive industry with free entry. If the inputs to the
industry are in perfectly elastic supply so that their price does not
depend on industry output, the (constant-cost) industry's long-run
supply curve is S on Figure 13-10--a horizontal line at price =
marginal cost = minimum average cost. If the price of some of the
inputs rises as the industry purchases more of them (an
increasing-cost industry), the result is an upward-sloped supply
curve, such as S'.

Two possible long-run supply curves for the batten
industry. S, which is horizontal at a price equal to minimum average
cost, is drawn on the assumption that inputs are available in
perfectly elastic supply. S' is drawn on the assumption that as
quantity increases, input prices are bid up.

Part 1 vs Part 2--Two Approaches
Compared

The short-run supply curve tells us how the firm
will respond to changes in price over periods too short to make it
worth changing the size of its factory; the long-run supply curve
tells how the firm will respond to what it regards as permanent
changes in price. We have now solved for both. In doing so, what have
we learned that we did not already know?

The most important lesson is how to calculate the
behavior of the firm over the short run. In all of the earlier
examples of this chapter, the firms had simple all-or-none patterns
of production. A widget factory either produced at capacity or shut
down; a ship continued to carry a full load of freight as long as it
got anything at all for doing so. We were, in effect, assuming the
cost curves shown in Figures 13-11a and 13-11b--marginal cost
constant up to some maximum level of production and infinite beyond
that. We were also assuming that there was only one kind of factory
and one kind of ship.

In analyzing the batten factory, we allowed for
more realistic cost curves. By doing so, we saw how, even in the
short run, quantity supplied can vary continuously with price. We
could have done the same thing in the earlier analysis; I chose not
to. All-or-none production was a simplifying assumption used to avoid
complications that were, at that point, inessential. The discussion
of long-run and short-run supply curves was a convenient point at
which to drop that simplification.

What are the disadvantages of the
short-run/long-run approach? One of them is that it encourages
students to confuse sunk costs and fixed costs. In the examples that
are used, the two are generally the same, but there is no reason why
they have to be.

In the batten industry, as I pointed out earlier,
the curve labeled average variable cost could also have been labeled
average recoverable cost, since the two are equal. I labeled it AVC
in deference to convention; that is how you will generally see it in
other textbooks. It would have been more correct to have labeled it
ARC. It is the fact that the cost is recoverable, not the fact that
it is variable, that is essential to the way in which the curve is
related to the short-run supply curve. If we were considering a
situation in which variable cost and recoverable cost were not the
same, we could have simply drawn the ARC curve and forgotten about
AVC.

One of the faults of the short-run/long-run
approach is that it encourages confusion between fixed and sunk
costs. One of its limitations is that it distinguishes between only
two kinds of costs--short-run and long-run. The more general approach
to sunk cost, which we developed earlier in the chapter, can be used
to analyze a much broader range of situations, including ones in
which there are several long-lived productive assets with different
lifetimes.

A second limitation is that the short-run/long-run
approach says nothing about what happens to price between the two
periods--how it adjusts over time to unexpected changes in demand. If
we know how many factories of what size exist, the short-run supply
curve allows us to calculate price and quantity; whether or not we
know how many factories exist, the long-run supply curve tells us
what price and quantity must eventually be if the situation remains
stable for long enough. But the approach does not explain how to
calculate the path by which price and quantity move from the one
value to the other--which is one of the things we did in analyzing
the widget and shipping industries.

None of this means that the short-run/long-run
approach is wrong. Both in using economics and in teaching it, one
must balance the costs and benefits of different degrees of
simplicity. The short-run/long-run approach described in this section
has the advantages and the disadvantages of greater simplicity; it is
easier to teach but tells us less of what we want to know than the
approach used earlier in the chapter.

In one sense, the difference is entirely
pedagogical. Once you understand either approach, you can develop the
other out of it. Starting with short- and long-run cost curves, you
could, with a little ingenuity, figure out how to analyze more
complicated cases or how to trace the path of price and quantity over
time. Starting with sunk costs, you can work out short-run and
long-run cost curves as special cases--not only in the shipping
industry of Figures 13-l through 13-4 but in more complicated
situations as well. By teaching the material in both ways, I hope I
have allowed you to learn it in whichever way you found more natural.
That is a benefit. Its cost is measured in additional pages of book
and additional hours of time--mine in writing and yours in reading.
The production of textbooks involves the same sort of trade-off
between costs and benefits as does the production of anything
else--or any other action requiring choice.

PART 3 - SPECULATION

It is difficult to read either newspapers or
history books without occasionally coming across the villainous
speculators. Speculators, it sometimes seems, are responsible for all
the problems of the world--famines, currency crises, high
prices.

How Speculation Works

A speculator buys things when he thinks they are
cheap and sells them when he thinks they are expensive. Imagine, for
example, that you decide there is going to be a bad harvest this
year. If you are right, the price of grain will go up. So you buy
grain now, while it is still cheap. If you are right, the harvest is
bad, the price of grain goes up, and you sell at a large
profit.

There are several reasons why this particular way
of making a profit gets so much bad press. For one thing, the
speculator is, in this case at least, profiting by other people's bad
fortune, making money from, in Kipling's phrase, "Man's belly pinch
and need." Of course, the same might be said of farmers, who are
usually considered good guys. For another, the speculator's purchase
of grain tends to drive up the price, making it seem as if he is
responsible for the scarcity.

But in order to make money, the speculator must
sell as well as buy. If he buys when grain is plentiful, he does
indeed tend to increase the price then; but if he sells when it is
scarce (which is what he wants to do in order to make money), he
increases the supply and decreases the price just when the additional
grain is most useful.

A different way of putting it is to say that the
speculator, acting for his own selfish motives, does almost exactly
what a benevolent despot would do. When he foresees a future scarcity
of wheat, he induces consumers to use less wheat now. The speculator
gets consumers to use less wheat now by buying it (before the
consumers themselves realize the harvest is going to be bad), driving
up the price; the higher price encourages consumers to consume less
food (by slaughtering meat animals early, for example, to save their
feed for human consumption), to import food from abroad, to produce
other kinds of food (go fishing, dry fruit, . . .), and in other ways
to prepare for the anticipated shortage. He then stores the wheat and
distributes it (for a price) at the peak of the famine. Not only does
he not cause famines, he prevents them.

More generally, speculators (in many things, not
just food) tend, if successful, to smooth out price movements, buying
goods when they are below their long-run price and selling them when
they are above it, raising the price towards equilibrium in the one
case and lowering it towards equilibrium in the other. They do what
governmental "price-stabilization" schemes claim to do--reduce
short-run fluctuations in prices. In the process, they frequently
interfere with such price-stabilization schemes, most of which are
run by producing countries and designed to "stabilize" prices as high
as possible.

Cui Bono

Why indeed should we welcome you, Master Stormcrow? Lathspell I
name you, ill-news; and ill news is an ill guest they say.

--Grima to Gandalf in The Two Towers by J.R.R. Tolkien

At least part of the unpopularity of speculators
and speculation may reflect the traditional hostility to bearers of
bad news; speculators who drive prices up now in anticipation of a
future bad harvest are conveying the fact of future scarcity and are
forcing consumers to take account of it. Part also may be due to the
difficulty of understanding just how speculation works. Whatever the
reason, ideas kill, and the idea that speculators cause shortages
must be one of the most lethal errors in history. If speculation is
unpopular it is also difficult, since the speculator depends for his
profit on not having his stocks of grain seized by mob or government.
In poor countries, which means almost everywhere through almost all
of history, the alternative to speculation in food crops is periodic
famine.

One reason people suspect speculators of causing
price fluctuations is summarized in the Latin phrase cui bono; a
loose translation would be "Who benefits?" If the newspapers discover
that a gubernatorial candidate has been receiving large campaign
donations from a firm that made $10 million off state contracts last
year, it is a fair guess that the information was fed to them by his
opponent. If a coup occurs somewhere in the Third World and the
winners immediately ally themselves with the Soviet Union (or the
United States), we do not have to look at the new ruler's bank
records to suspect that the takeover was subsidized by Moscow (or
Washington).

While cui bono is a useful rule for understanding
many things, it is not merely useless but positively deceptive for
understanding price movements. The reason is simple. The people who
benefit from an increase in the price of something are those who
produce it, but by producing, they drive the price not up but down.
The people who benefit by a price drop are those who buy and consume
the good, but buying a good tends to increase its price, not lower
it. The manufacturer of widgets may spend his evenings on his knees
praying for the price of widgets to go up, but he spends his days
behind a desk making it go down. Hence the belief that price changes
are the work of those who benefit by them is usually an error and
sometimes a dangerous one.

Speculators make money by correctly predicting
price changes, especially those changes that are difficult to
predict. It is natural enough to conclude, according to the principle
of cui bono, that speculators cause price fluctuations.

The trouble with this argument is that in order to
make money, a speculator must buy when prices are low and sell when
they are high. Buying when prices are low raises low prices; selling
when prices are high lowers high prices. Successful speculators
decrease price fluctuations, just as successful widget makers
decrease the price of widgets. Destabilizing speculators are, of
course, a logical possibility; they can be recognized by the red ink
in their ledgers. The Hunt brothers of Texas are a notable recent
example. A few years ago, they lost several billion dollars in the
process of driving the price of silver up to what turned out to be
several times its long-run equilibrium level.

It is true, of course, that a speculator would
like to cause instability, supposing that he could do so without
losing money; more precisely, he would like to make the prices of
things he is going to sell go up before he sells them and of things
he is going to buy go down before he buys them. He cannot do this by
his market activities, but he can try to spread misleading rumors
among other speculators; and, no doubt, some speculators do so. His
behavior in this respect is like that of a producer who advertises
his product; he is trying to persuade people to buy what he wants to
sell. The speculator faces an even more skeptical audience than the
advertiser, since it is fairly obvious that if he really expected the
good to go up he would keep quiet and buy it himself. So the private
generating of disinformation, while it undoubtedly occurs, is
unlikely to be very effective.

I once heard a talk by an economist who had
applied the relationship between stabilization and profitable
speculation in reverse. The usual argument is that speculators, by
trying to make a profit, provide the useful public service of
stabilizing prices. The reverse argument involved not private
speculators but central banks. Central banks buy and sell currencies,
supposedly in order to stabilize exchange rates (an exchange rate is
the price of one kind of money measured in another). They are widely
suspected (by economists and speculators) of trying to keep exchange
rates not stable but above or below their market clearing
levels.

If profitable speculation is stabilizing, one
might expect successful stabilization of currencies to be profitable.
If the banks are buying dollars when they are temporarily cheap and
selling them when they are temporarily expensive, they should be both
stabilizing the value of the dollar and making a profit. One
implication of this argument is that the central banks are
superfluous--if there are profits to be made by stabilizing
currencies, speculators will be glad to volunteer for the job. A
second implication is that we can judge the success of central banks
by seeing whether they in fact make or lose money on their
speculations. The conclusion of the speaker, who had studied
precisely that question, was that they generally lost money.

OPTIONAL SECTION

CHOICE IN AN UNCERTAIN WORLD

In Chapters 1-11, we saw how markets work to
determine prices and quantities in a certain and unchanging world. In
Chapter 12, we learned how to deal with a world that was changing but
certain. In such a world, any decision involves a predictable stream
of costs and benefits--so much this year, so much next year, so much
the year after. One simply converts each stream into its present
value and compares the present values of costs and benefits, just as
we earlier compared annual flows of costs and benefits.

The next step is to analyze individual choice in
an uncertain world. Again our objective is to convert the problem we
are dealing with into the easier problem we have already solved. To
describe an uncertain world, we assume that each individual has a
probability distribution over possible outcomes. He does not know
what will happen but he knows, or believes he knows, what might
happen and how likely it is to happen. His problem, given what he
knows, is how to achieve his objectives as successfully as
possible.

The Rational Gambler

Consider, for example, an individual betting on
whether a coin will come up heads or tails. Assuming the coin is a
fair one, half the time it will come up heads and half the time
tails. The gambler's problem is to decide what bets he should be
willing to take.

The answer seems obvious--take any bets that offer
a payoff of more than $1 for each $1 bet; refuse any that offer less.
If someone offers to pay you $2 if the coin comes up heads, on
condition that you pay him $1 if it comes up tails, then on average
you gain by accepting the bet and should do so. If he offers you
$0.50 for the risk of $1, then on average you lose by accepting; you
should refuse the bet.

In these examples, you are choosing between a
certain outcome (decline the bet--and end up with as much money as
you started with) and an uncertain outcome (accept the bet--end up
with either more or less). A more general way of putting the rule is
that in choosing among alternatives, you should choose the one that
gives you the highest expected return, where the expected return is
the sum of the returns associated with the different possible
outcomes, each weighted by its probability.

Maximizing Expected Return. This is the correct
answer in some situations but not in all. If you make a fifty-fifty
bet many times, you are almost certain to win about half the time; a
bet that on average benefits you is almost certain to give you a net
gain in the long term. If, for instance, you flip a fair coin 1,000
times, there is only a very small chance that it will come up heads
more than 600 times or fewer than 400. If you make $2 every time it
comes up heads and lose $1 every time it comes up tails, you are
almost certain, after 1,000 flips, to be at least $200 ahead.

The case of the gambler who expects to bet many
times on the fall of a coin can easily be generalized to describe any
game of chance. The rule for such a gambler is "Maximize expected
return." Since we defined expected return as the sum, over all of the
possible outcomes, of the return from each outcome times the
probability of that outcome, we have:

<R>
piRi. (Equation 1)

Here pi is the probability of
outcome number i occurring, Ri is the return from outcome
number i, and <R> is the expected return.

When you flip a coin, it must come up either heads
or tails; more generally, any gamble ends up with some one of the
alternative outcomes happening, so we have:

pi = 1. (Equation 2)

In the gamble described earlier, where the gambler
loses $1 on tails and gains $2 on heads, we have:

Here p1 and p2, the probabilities of heads
and tails respectively, are each equal to one half; your expected
return is $0.50. If you play the game many times, you will on average
make $0.50 each time you play. The expected return from taking the
gamble is positive, so you should take it--provided you can repeat it
many times. The same applies to any other gamble with a positive
expected return. A gamble with an expected return of zero--you are on
average equally well off whether or not you choose to take it--is
called a fair gamble.

We now know how a gambler who will take the same
gamble many times should behave. In choosing among several gambles,
he should take the one with the highest expected return. In the
particular case where he is accepting or declining bets, so that one
of his alternatives is a certainty of no change, he should take any
bet that is better than a fair gamble.

Maximizing Expected Utility. Suppose, however,
that you are only playing the game once--and that the bet is not $1
but $50,000. If you lose, you are destitute--$50,000 is all you have.
If you win, you gain $100,000. You may feel that a decline in your
wealth from $50,000 to zero hurts you more than an increase from
$50,000 to $150,000 helps you. One could easily enough imagine
situations in which losing $50,000 resulted in your starving to death
while gaining $100,000 produced only a modest increase in your
welfare.

Such a situation is an example of what we earlier
called declining marginal utility. The dollars that raise you from
zero to $50,000 are worth more per dollar than the additional dollars
beyond $50,000. That is precisely what we would expect from the
discussion of Chapter 4. Dollars are used to buy goods; we expect
goods to be worth less to you the more of them you have.

When you choose a profession, start a business,
buy a house, or stake your life savings playing the commodity market,
you are betting a large sum, and the bet is not one you will repeat
enough times to be confident of getting an average return. How can we
analyze rational behavior in such situations?

The answer to this question was provided by John
Von Neumann, the same mathematician mentioned in Chapter 11 as the
inventor of game theory. He demonstrated that by combining the idea
of expected return used in the mathematical theory of gambling
(probability theory) with the idea of utility used in economics, it
was possible to describe the behavior of individuals dealing with
uncertain situations--whether or not the situations were repeated
many times.

The fundamental idea is that instead of maximizing
expected return in dollars, as in the case described above,
individuals maximize expected return in utiles--expected utility.
Each outcome i has a utility Ui. We define expected utility
as:

<U> piUi. (Equation 3)

Your utility depends on many things, of which the
amount of money you have is only one. If we are considering
alternatives that only differ with regard to the amount of money you
end up with, we can write:

Ui = U(Ri).

Or, in other words, the utility you get from
outcome i depends only on how much more (or less) money that outcome
gives you. If utility increases linearly with income, as shown on
Figure 13-12, we have:

U(R) = A + (B x R);

<U> =
piUi
= pi(A + BRi) =A pi + B
piRi
=

= A + B<R>. (Equation 4)

Comparing the left and right-hand sides of
Equation 4, we see that whatever decision maximizes <R> also
maximizes <U>. In this case--with a linear utility
function--the individual maximizing his expected utility behaves like
the gambler maximizing his expected return.

A Methodological Digression. In going from
gambling games to utility graphs, we have changed somewhat the way in
which we look at expected return. In the case of gambling, return was
defined relative to your initial situation--positive if you gained
and negative if you lost. That was a convenient way of looking at
gambling because the gambler always has the alternative of refusing
to bet and so ending up with a return of zero. But in an uncertain
world, the individual does not usually have that alternative;
sometimes--indeed almost always--he is choosing among alternatives
all of which are uncertain. In that context, it is easier to define
zero return as ending up with no money at all and to measure all
other outcomes relative to that. We can then show the utility of any
outcome on a graph such as Figure 13-12 as the utility of the income
associated with that outcome. If you start with $10,000 and bet all
of it at even odds on the flip of a coin--heads you win, tails you
lose--then the utility to you of the outcome "heads" is the utility
of $20,000. The utility to you of the outcome "tails" is the utility
of zero dollars.

A second difficulty with Figure 13-12 is the ambiguity as to just
what is being graphed on the horizontal axis--what is utility a
function of? Is it income (dollars per year) or money (dollars)?
Strictly speaking, utility is a flow (utiles per year) that depends
on a flow of consumption (apples per year). The utility we get by
consuming 100 apples, or whatever else we buy with our income,
depends in part on the period of time over which we consume
them.

If I were being precise, I would do all the
analysis in terms of flows and compare alternatives by comparing the
present values of those flows, in dollars or utiles. This would make
the discussion a good deal more complicated without adding much to
its content. It is easier to think of Figure 13-12, and similar
figures, as describing either someone who starts with a fixed amount
of money and is only going to live for a year, or, alternatively,
someone with a portfolio of bonds yielding a fixed income who is
considering gambles that will affect the size of his portfolio. The
logic of the two situations is the same. In the one case, the figure
graphs the utility flow from a year's expenditure; in the other case,
it graphs the present value of the utility flow from spending the
same amount every year forever. Both approaches allow us to analyze
the implications of uncertainty while temporarily ignoring other
complications of a changing world. To make the discussion simpler, I
will talk as if we are considering the first case; that way I can
talk in "dollars" and "utiles" instead of "dollars per year" and
"utiles per year." The amount of money you have may still sometimes
be described as your income--an income of x dollars/year for one year
equals x dollars.

Risk Preference and the Utility
Function

Or

As I Was Saying When I So Rudely
Interrupted Myself

Figure 13-12 showed utility as a linear function
of income; Figure 13-13a shows a more plausible relation. This time,
income has declining marginal utility. Total utility increases with
income, but it increases more and more slowly as income gets higher
and higher.

Suppose you presently have $20,000 and have an
opportunity to bet $10,000 on the flip of a coin at even odds. If you
win, you end up with $30,000; if you lose, you end up with
$10,000.

In deciding whether to take the bet, you are
choosing between two different gambles. The first, the one you get if
you do not take the bet, is a very simple gamble indeed--a certainty
of ending up with $20,000. The second, the one you get if you do take
the bet, is a little more complicated--a 0.5 chance of ending up with
$10,000 and a 0.5 chance of ending up with $30,000. So for the first
gamble, we have:

The individual takes the alternative with the
higher expected utility; he declines the bet. In money terms, the two
alternatives are equally attractive; they have the same expected
return. In that sense, it is a fair bet. In utility terms, the first
alternative is superior to the second. You should be able to convince
yourself that as long as the utility function has the shape shown in
Figure 13-13a, an individual will always prefer a certainty of $1 to
a gamble whose expected return is $1.

An individual who behaves in that way is risk
averse. A utility function that is almost straight, such as Figure
13-13b, represents an individual who is only slightly risk averse.
Such an individual would decline a fair gamble but might accept one
that was a little better than fair--bet $1,000 against $1,100 on the
flip of a coin, for example. An individual who was extremely risk
averse (Figure 13-13c) might still accept a gamble--but only one with
a very high expected return, such as risking $1,000 on the flip of a
coin to get $10,000.

Total utility of income for a risk-averse
individual. Figure 13-13b corresponds to
an individual who is only slightly risk averse; he will refuse a fair
gamble but accept one that is slightly better than fair. Figure
13-13c corresponds to an individual who is very risk averse; he will
accept a gamble only if it is much better than a fair gamble.

Figure 13-14a shows the utility function of a risk
preferrer. It exhibits increasing marginal utility. A risk preferrer
would be willing to take a gamble that was slightly worse than
fair--although he would still decline one with a sufficiently low
expected return. An individual who is neither a risk preferrer nor a
risk averter is called risk neutral. The corresponding utility
function has already been shown--as Figure 13-12.

Consider an individual who requires a certain
amount of money in order to buy enough food to stay alive. Increases
in income below that point extend his life a little and so are of
some value to him, but he still ends up starving to death. An
increase in income that gives him enough to survive is worth a great
deal to him. Once he is well past that point, additional income buys
less important things, so marginal utility of income falls. The
corresponding utility function is shown as Figure 13-14b; marginal
utility first rises with increasing income, then falls.

Such an individual would be a risk preferrer if
his initial income were at point A, below subsistence. He would be a
risk averter if he were starting at point B. In the former case, he
would, if necessary, risk $1,000 to get $500 at even odds. If he
loses, he only starves a little faster; if he wins, he lives.

In discussing questions of this sort, it is
important to realize that the degree to which someone exhibits risk
preference or risk aversion depends on three different things--the
shape of his utility function, his initial income, and the size of
the bet he is considering. For small bets, we would expect everyone
to be roughly risk neutral; the marginal utility of a dollar does not
change very much between an income of $19,999 and an income of
$20,001, which is the relevant consideration for someone with $20,000
who is considering a $1 bet.

The Simple Cases. The expected return from a
gamble depends only on the odds and the payoffs; the expected utility
depends also on the tastes of the gambler, as described by his
utility function. So it is easier to predict the behavior of someone
maximizing his expected return than of someone maximizing expected
utility. This raises an interesting question--under what
circumstances are the two maximizations equivalent? When does someone
maximize his utility by maximizing his expected return?

Total utility of income for a risk preferrer
and for someone who is risk preferring for some incomes and risk
averse for others. Figure 13-14b shows
total utility of income for someone who requires about $1,500 to stay
alive. Below that point, the marginal utility of income (the slope of
total utility) increases with increasing income; above that point, it
decreases.

We saw one answer at the beginning of this section
of the chapter. An individual who makes the same gamble many times
can expect the results to average out. In the long run, the outcome
is almost certain--he will get something very close to the expected
value of the gamble times the number of times he takes it. Since his
income at the end of the process is (almost) certain, all he has to
do in order to maximize his expected utility is to make that income
as large as possible--which he does by choosing the gamble with the
highest expected return.

There are three other important situations in
which maximizing expected utility turns out to be equivalent to
maximizing expected return. One is when the individual is
risk-neutral, as shown on Figure 13-12. A second is when the size of
the prospective gains and losses is small compared to one's income.
If we consider only small changes in income, we can treat the
marginal utility of income as constant; if the marginal utility of
income is constant, then changes in utility are simply proportional
to changes in income, so whatever choice maximizes expected return
also maximizes expected utility.

One can see the same thing geometrically. Figure
13-15 is a magnified version of part of Figure 13-13a. If we consider
only a very small range of income--between $19,900 and $20,000, for
instance--the utility function is almost straight. For a
straight-line utility function, as I showed earlier, maximizing
expected utility is equivalent to maximizing expected return. So if
we are considering only small changes in income, we should act as if
we were risk neutral.

Magnified version of part of Figure
13-13a. Although the total utility curve
shown on Figure 13-13a is curved, corresponding to risk aversion, any
small section of it appears almost straight. This corresponds to the
fact that the marginal utility of income is almost constant over
small ranges of income; individuals are almost risk neutral for small
gambles.

Next consider the case of a corporation that is
trying to maximize the market value of its stock--as the discussion
of takeover bids in the optional section of Chapter 9 suggests that
corporations tend to do. In an uncertain world, what management is
really choosing each time it makes a decision is a probability
distribution for future profits. When the future arrives and it
becomes clear which of the possible outcomes has actually happened,
the price of the stock will reflect what the profits actually are. So
in choosing a probability distribution for future profits, management
is also choosing a probability distribution for the future price of
the stock.

How is the current market value of a stock related
to the probability distribution of its future value? That is a
complicated question--one that occupies a good deal of the theory of
financial markets; if you are an economics major, you will probably
encounter it again. The short, but not entirely correct, answer is
that the current price of the stock is the expected value of the
future price--the average over all possible futures weighted by the
probability of each. The reason is that the buyer of stock is in the
same position as the gambler discussed earlier; he can average out
his risks by buying a little stock in each of a large number of
companies. If he does, his actual return will be very close to his
expected return. If the price of any particular stock were
significantly lower than the expected value of its future price,
investors would all want to buy some of it; if the price were higher
than the expected value of its future price, they would all want to
sell some. The resulting market pressures force the current price
toward the expected value of future prices.

If, as suggested above, management wishes to
maximize the present price of its stock, it must try to maximize the
expected value of its future price. It does that by maximizing the
expected value of future profits. So it acts like the gambler we
started with; it maximizes expected returns.

This is true only if the firm is trying to
maximize the value of its stock. The threat of takeover bids has some
tendency to make it do so. It is not clear how strong that tendency
is--how closely that threat constrains management. To the extent that
management succeeds in pursuing its own goals rather than those of
the stockholders, the conclusion no longer holds. If the firm takes a
risk and goes bankrupt, the (present and future) income of the chief
executive may fall dramatically. If so, he may well be unwilling to
make a decision that has a 50 percent chance of leading to bankruptcy
even if it also has a 50 percent chance of tripling the firm's
value.

Insurance. The existence of individuals who are
risk averse provides one explanation for the existence of insurance.
Suppose you have the utility function shown in Figure 13-13a. Your
income is $20,000, but there is a small probability --0.01--of some
accident that would reduce it to $10,000. The insurance company
offers to insure you against that accident for a price of $100.
Whether or not the accident happens, you give them $100. If the
accident happens, they give you back $10,000. You now have a choice
between two gambles--buying or not buying insurance. If you buy the
insurance, then, whether or not the accident occurs, the outcome is
the same--you have $20,000 minus the $100 you paid for the insurance
(I assume the accident only affects your income). So for the first
gamble, you have:

You are better off with the insurance than without
it, so you buy the insurance.

In the example as given, the expected
return--measured in dollars--from buying the insurance was the same
as the expected return from not buying it. Buying insurance was a
fair gamble--you paid $100 in exchange for one chance in a hundred of
receiving $10,000. The insurance company makes hundreds of thousands
of such bets, so it will end up receiving, on average, almost exactly
the expected return. If insurance is a fair gamble, the money coming
in to buy insurance exactly balances the money going out to pay
claims. The insurance company neither makes nor loses money; the
client breaks even in money but gains in utility.

Insurance companies in the real world have
expenses other than paying out claims--rent on their offices,
commissions to their salespeople, and salaries for their
administrators, claim investigators, adjusters, and lawyers. In order
for an insurance company to cover all its expenses, the gamble it
offers must be somewhat better than a fair one from its standpoint.
If so, it is somewhat worse than fair from the standpoint of the
company's clients.

The clients may still find that it is in their
interest to accept the gamble and buy the insurance. If they are
sufficiently risk averse, an insurance contract that lowers their
expected return may still increase their expected utility. In the
case discussed above, for example, it would still be worth buying the
insurance even if the company charged $130 for it. It would not be
worth buying at $140. You should be able to check those results for
yourself by redoing the calculations that showed that the insurance
was worth buying at $100.

Earlier I pointed out that with regard to risks
that involve only small changes in income, everyone is (almost) risk
neutral. One implication of this is that it is only worth insuring
against large losses. Insurance is worse than a fair gamble from the
standpoint of the customer, since the insurance company has to make
enough to cover its expenses. For small losses, the difference
between the marginal utility of income before and after the loss is
not large enough to convert a loss in expected return into a gain in
expected utility.

The Lottery-Insurance Puzzle. Buying a ticket in a
lottery is the opposite of buying insurance. When you buy insurance,
you accept an unfair gamble--a gamble that results, on average, in
your having less money than if you had not accepted it--in order to
reduce uncertainty. When you buy a lottery ticket, you also accept an
unfair gamble--on average, the lottery pays out in prizes less than
it takes in--but this time you do it in order to increase your
uncertainty. If you are risk averse, it may make sense for you to buy
insurance--but you should never buy lottery tickets. If you are a
risk preferrer it may make sense for you to buy a lottery ticket--but
you should never buy insurance.

This brings us to a puzzle that has bothered
economists for at least 200 years--the lottery-insurance paradox. In
the real world, the same people sometimes buy both insurance and
lottery tickets. Some people both gamble when they know the odds are
against them and buy insurance when they know the odds are against
them. Can this be consistent with rational behavior?

There are at least two possible ways in which it
can. One is illustrated on Figure 13-16. The individual with the
utility function shown there is risk averse for one range of incomes
and risk preferring for another, higher, range. If he starts at point
A, in between the two regions, he may be interested in buying both
insurance and lottery tickets. Insurance protects him against risks
that move his income below A--where he is risk averse. Lottery
tickets offer him the possibility (if he wins) of an income above
A--where he is risk preferring.

This solution is logically possible, but it does
not seem very plausible. Why should people have such peculiarly
shaped utility functions, with the value to them of an additional
dollar first falling with increasing income then rising again? And if
they do, why should their incomes just happen to be near the border
between the two regions?

Another explanation of the paradox is that in the
real-world situation we observe, one of the conditions for our
analysis does not hold. So far, we have been considering situations
where the only important difference among the outcomes is money; the
utility of each outcome depends only on the amount of money it leaves
you with. It is not clear that this is true for the individuals who
actually buy lottery tickets.

One solution to the lottery-insurance puzzle. The
total utility function shows declining marginal utility of income
(risk aversion) to the left of point A and increasing marginal
utility of income (risk preference) to the right. An individual at A
may increase his expected utility by buying both insurance and
lottery tickets.

Consider the lotteries you have yourself been
offered--by Reader's Digest, Publisher's Clearinghouse, and similar
enterprises. The price is the price of a stamp, the payoff--lavishly
illustrated with glossy photographs--a (very small) chance of a new
Cadillac, a Caribbean vacation, an income of $20,000 a year for life.
My rough calculations--based on a guess of how many people respond to
the lottery--suggest that the value of the prize multiplied by the
chance of getting it comes to less than the cost of the stamp. The
expected return is negative.

Why then do so many people enter? The explanation
I find most plausible is that what they are getting for their stamp
is not merely one chance in a million of a $40,000 car. They are also
getting a certainty of being able to daydream about getting the
car--or the vacation or the income--from the time they send in the
envelope until the winners are announced. The daydream is made more
satisfying by the knowledge that there is a chance, even if a slim
one, that they will actually win the prize. The lottery is not only
selling a gamble. It is also selling a dream--and at a very low
price.

This explanation has the disadvantage of pushing
such lotteries out of the area where economics can say much about
them; we know a good deal about rational gambling but very little
about the market for dreams. It has the advantage of explaining not
only the existence of lotteries but some of their characteristics. If
lotteries exist to provide people a chance of money, why do the
prizes often take other forms; why not give the winner $40,000 and
let him decide whether to buy a Cadillac with it? That would not only
improve the prize from the standpoint of the winner but would also
save the sponsors the cost of all those glossy photographs of the
prizes.

But many people may find it easier to daydream
about their winnings if the winnings take a concrete form. So the
sponsors (sometimes) make the prizes goods instead of money--and
provide a wide variety of prizes to suit different tastes in
daydreams. This seems to be especially true of "free" lotteries--ones
where the price is a stamp and the sponsor pays for the prizes out of
someone's advertising budget instead of out of ticket receipts.
Lotteries that sell tickets seem more inclined to pay off in
money--why I do not know.

In Chapter 1, I included in my definition of
economics the assumption that individuals have reasonably simple
objectives. You will have to decide for yourself whether a taste for
daydreams is consistent with that assumption. If not, then we may
have finally found something that is not an economic question--as
demonstrated by our inability to use economics to answer it.

Von Neumann Utility

Near the beginning of this section, I said that
John Von Neumann was responsible for combining the ideas of utility
and choice under uncertainty. So far, I have shown how the two ideas
are combined but have said very little about exactly what Von Neumann
(in conjunction with economist Oskar Morgenstern) contributed. You
may reasonably have concluded that the great idea was simply to
assert "People maximize expected utility" and keep talking--in the
hope that nobody would ask "Why?"

What Von Neumann and Morgenstern actually did was
both more difficult and more subtle than that. They proved that if
you assume that individual choice under uncertainty meets a few
simple consistency conditions, it is always possible to assign
utilities to outcomes in such a way that the decisions people
actually make are the ones they would make if they were maximizing
expected utility.

Von Neumann and Morgenstern start by considering
an individual choosing among "lotteries." A lottery is a collection
of outcomes, each with a probability. Some outcome must occur, so all
the probabilities together add up to one. Just as, in considering
ordinary utility functions, we assume that the individual can choose
between any two bundles, so they assumed that given any two lotteries
L and M, the individual either prefers L to M, prefers M to L, or is
indifferent between them. They further assumed that preferences are
transitive; if you prefer L to M and M to N, you must prefer L to
N.

Another assumption was that in considering
lotteries whose payoffs are themselves lotteries--probabilistic
situations whose outcomes are themselves probabilistic
situations--people combine probabilities in a mathematically correct
fashion. If someone is offered a ticket giving him a 50 percent
chance of winning a lottery ticket, which in turn gives him a 50
percent chance of winning a prize, he regards that compound lottery
as equivalent to a ticket giving a 25 percent chance of winning the
same prize--and similarly for any other combination of
probabilities.

The remaining two assumptions involve the
continuity of preferences. One is that if I prefer outcome A to
outcome B, I also prefer to B any lottery that gives me some
probability of getting A and guarantees that if I do not get A, I
will get B. The final assumption is that if I prefer outcome A to
outcome B and outcome B to outcome C, then there is some probability
mix of A and C--some lottery containing only those outcomes--that I
consider equivalent to B. To put it in different words, this says
that as I move from a certainty of A to a certainty of C via various
mixtures of the two, my utility changes continuously from U(A) to
U(C). Since by assumption U(A) > U(B) > U(C)--that is what the
"if" clause at the beginning of this paragraph says--as my utility
moves continuously from U(A) to U(C) it must at some intermediate
point be equal to U(B).

All of these assumptions seem reasonable as part
of a description of "rational" or "consistent" behavior under
uncertainty. If an individual's behavior satisfies them, it is
possible to define a Von Neumann utility function--a utility for
every outcome--such that the choices he actually makes are the
choices he would make if he were trying to maximize his expected
utility. That is what Von Neumann and Morgenstern proved.

In the optional section of Chapter 3, I pointed
out that utility as then defined contained a considerable element of
arbitrariness; utility functions were supposed to describe behavior,
but exactly the same behavior could be described by many different
utility functions. We could deduce from observing individuals'
choices that they preferred A to B, but not by how much. Even the
principle of declining marginal utility, to which I several times
referred, is, strictly speaking, meaningless in that context; if you
cannot measure the amount by which I prefer one alternative to
another, then you cannot say whether the additional utility that I
get when my income increases from $9,000/year to $10,000 is more or
less than when it increases from $10,000 to $11,000. Declining
marginal utility then has content only in the form of the declining
marginal rate of substitution--a concept that, as I pointed out at
the time, is closely related but not equivalent.

Once we accept the Von Neumann-Morgenstern
definition of utility under uncertainty, that problem vanishes. The
statement "I prefer outcome C to outcome B by twice as much as I
prefer B to A" is equivalent to "I am indifferent between a certainty
of B and a lottery that gives me a two-thirds chance of A and a
one-third chance of C."

To see that the two statements are equivalent, we
will work out the expected utilities for the two alternatives
described in the second statement and show that the first statement
implies that they are equal, as follows:

Let Lottery 1 consist of a certainty of B, Lottery
2 of a two-thirds chance of A and a one-third chance of C. We have
for Lottery 1:

p1 = 1; U1 = U(B); <U> = U(B).

We have for Lottery 2:

p1 = 2/3; U1 = U(A);

p2 = 1/3; U2 = U(C);

<U> = p1U1 + p2U2 = 2/3 U(A) + 1/3
U(C).

Statement 1 tells us that:

U(C) - U(B) = 2 x (U(B) - U(A)).

Rearranging this gives us:

U(C) + 2 x U(A) = 3 x U(B);

2/3 U(A) + 1/3 U(C) = U(B). (Equation 5)

The left-hand side of Equation 5 is the expected
utility of Lottery 2, and the right-hand side is the expected utility
of Lottery 1, so the expected utilities of the two alternatives are
the same; the individual is indifferent between them.

We have now shown that Statement 1 implies
Statement 2. We could equally well have started with Statement 2 and
worked backward to Statement 1. If each statement implies the other,
then they are equivalent.

So using utility functions to describe choice
among probabilistic alternatives makes the functions themselves
considerably less arbitrary. In our earlier discussion of utility,
the only meaningful statements were of the form "A has more utility
to me than B" or, equivalently, "I prefer A to B." Now the statement
"Going from A to B increases my utility by twice as much as going
from C to D" (or, equivalently, "I prefer A to B twice as much as I
prefer C to D") has meaning as well. If we can make quantitative
comparisons of utility differences, we can also make quantitative
comparisons of marginal utilities, so the principle of declining
marginal utility means something. We saw exactly what it meant a few
pages ago; the statement "My marginal utility for income is
declining" is equivalent to "I am risk averse." Similarly, the
statement "My marginal utility for ice cream cones is declining" is
equivalent to "I am risk averse if expected return is in ice cream
cones rather than in dollars. I would not accept a gamble that
consisted of a 50 percent chance of getting an ice cream cone and a
50 percent chance of losing one."

We have eliminated much of the arbitrariness from
utility functions but not all of it. Nothing we have done tells us
how big a utile is, so a change in scale is still possible. If I say
that I prefer A to B by 10 utiles, B to C by 5, and C to D by 2,
while you insist that the correct numbers are 20, 10, and 4, no
possible observation of my behavior could prove one of us right and
one wrong. We agree about the order of preferences; we agree about
their relative intensity--all we disagree about is the size of the
unit in which we are measuring them.

It is also true that nothing we have done tells us
where the zero of the utility function is. If I claim that my
utilities for outcomes A, B, and C are 0, 10, 30, while you claim
they are - 10, 0, and 20, there is again no way of settling the
disagreement. We agree about the order, we agree about the
differences--all we disagree about is which alternative has zero
utility. So changes in the utility function that consist of adding
the same amount to all utilities (changing the zero), or multiplying
all utilities by the same number (changing the scale), or both, do
not really change the utility function. The numbers are different,
but the behavior described is exactly the same. This means, for those
of you who happen to be mathematicians, that utility functions are
arbitrary with respect to linear transformations.

My own preference is to define zero as
nonexistence or death; that, after all, is the one outcome in which
one gets, so far as I know, neither pleasure nor pain. A friend and
colleague once commented to me that she was not certain whether the
present value of utility at birth was positive or negative--meaning
that she was not sure whether, on net, life was worth living. I
concluded that her life had been much harder than mine.

Buying Information

You have decided to buy a car and are choosing
between two alternatives: a Honda Accord and a Nissan Stanza. From
previous experience, you expect that you will like one of the cars
better than the other, but unfortunately you do not know which. If
forced to state your opinions more precisely, you would say that you
think your consumer surplus would be $500 higher if you bought the
better car, and the probability that the Accord is better than the
Stanza is exactly 0.5 .

You consider two strategies. You can randomly
choose one of the cars and buy it. Alternatively, you can rent an
Accord for your next long trip, a Stanza for the trip after that, and
then decide which to buy. You believe that after having driven each
car a substantial distance, you will know with certainty which you
like better. Since it is more expensive to rent a car than to use a
car you own, the second strategy will cost you an extra $200. Should
you do it?

The answer depends on your utility function. You
are choosing between two lotteries. The first has payoffs of $0 and
$500, each with probability 0.5. The second has a certain payoff of
$300, since you get the extra consumer surplus but pay $200 for it.
If you are risk neutral or risk averse, you prefer a certainty of
$300 to a 0.5 chance of $500, so you rent the cars before you buy. If
you are a strong risk preferrer, you prefer the gamble, so you buy
without renting.

This simple problem illustrates the general idea
of buying information. By paying some search cost you can reduce
uncertainty, improving, on average, the outcomes of your decisions.
To decide whether the search cost is worth paying, you compare
expected utility without search to expected utility with search,
remembering to include the cost of the search in your
calculation.

In this particular case you had only two
alternatives, to search or not to search, and searching gave you
complete information--you knew with certainty which car you
preferred. In more general cases you may have to decide just how much
searching to do; the more you search, the better your information.
The correct rule is to search up to the point where the value of the
marginal increase in your expected utility from searching a little
more is just equal to the cost.

One example of such behavior that has received a
great deal of attention is the problem of job search. Many people who
consider themselves unemployed could find a job almost instantly--if
they were willing to wait on tables, or wash dishes, or drive a cab.
What they are looking for is not a job but a good job. The longer
they look, the better, on average, will be the best job opportunity
they find. Their rational strategy is to keep looking as long as they
expect to gain more from additional search than it costs them. Such
search unemployment makes up a significant fraction of the measured
unemployment rate.

One implication of this is that increases in
unemployment compensation tend to increase the unemployment rate. The
reason is not that the unemployed are lazy bums who prefer collecting
unemployment to working, but that they are rational searchers. The
higher the level of unemployment compensation is, the lower the cost
of being unemployed while searching for a job. The less it costs to
search, the more searching it pays to do.

Issues associated with acquiring and using
information provide some of the most interesting and difficult
questions in economics. They first appeared back in Chapter 1, where
I briefly mentioned the problem of incorporating information costs
into the definition of rationality, and will reappear in Chapter
18.

Where We Are Now

In the first 11 chapters of this book, we used
economics to understand how markets work in a certain and unchanging
world. It may have occurred to you that doing so was a waste of time,
since we live in a world that is uncertain and changing.

Looking back at what we have done in Chapters 12
and 13, you may now see why the book is organized in this way. In
Chapter 12, we learned how to analyze choice in a changing (but
certain) world using the same tools developed for an unchanging
world--simply evaluate costs and benefits in terms of present values
instead of annual flows. Now we have learned how to analyze choice in
an uncertain world by again using the same tools; we merely evaluate
costs and benefits by comparing the expected utilities of
probabilistic outcomes instead of the utilities of certain outcomes.
Combining the lessons of the two chapters in order to analyze choice
in a world that is both changing and uncertain would be
straightforward--evaluate choices in terms of the present value of
expected utility.

What we have done is to first solve economics in a
simple world and then show that the more complicated and realistic
world can, for purposes of economic analysis, be reduced to the
simple one. Introducing time and change does create some new
problems, such as those associated with sunk costs. Yet it is still
true that in learning to deal with the simple world of Chapters 1-11
we learned most of the basic ideas of economics, and that in Chapters
12 and 13 we have taken a large step towards making those ideas
applicable to the world we live in.

A Philosophical Digression

The concept of utility originated during the
nineteenth century among thinkers interested in both philosophy and
economics. It was proposed as an answer to the question "What should
a society maximize?" The utilitarians asserted that a society should
be designed to maximize the total utility of its members.

Their position has been heavily criticized over
the years and is now in poor repute among philosophers. One of the
major criticisms was that although we can, in principle, determine
whether you prefer A to B by more than you prefer C to D, there seems
to be no way of determining whether I prefer A to B by more than you
prefer C to D. There is no way of making interpersonal comparisons of
utility, no way of deciding whether a change that benefits me (gives
me A instead of B) and injures you (gives you D instead of C)
increases or decreases total utility.

One possible reply to this criticism of
utilitarianism goes as follows. Suppose we define utility in the
sense of Von Neumann and Morgenstern and use it to evaluate some
question such as "Should the United States abolish all tariffs?" It
turns out that the utilitarian rule--"Maximize total utility"--is
equivalent to another rule that some find intuitively more
persuasive: "Choose that alternative you would prefer if you knew you
were going to be one of the people affected but had no idea
which."

Why are the two equivalent? If I have no idea who
I am going to be, I presumably have an equal probability p of being
each person; if there are N people involved, then p = 1/N. If we
write the utility of person i as Ui, then the lottery that
consists of a probability p of being each person has an expected
utility:

<U> =
piUi
= pUi, = p Ui.

But
Ui is simply
the total utility of the society, so whichever alternative maximizes
total utility also maximizes <U>.

PROBLEMS

1. How should the developers of a new airliner
take account of the plane's design costs in deciding whether to
design and build the plane? In determining the price to charge
airline companies? Should they suspend production if they find that
they cannot obtain a price that will cover design costs?

2. After reading this chapter, you are considering
dropping this course. What costs should you take into account in
deciding whether to do so? What costs that you should ignore in that
decision should you have taken into account in deciding to take the
course in the first place?

3. Figure 13-17a shows the cost curves for
producing typewriters in a typewriter factory. The inputs are
available in perfectly elastic supply; all firms are identical and
there are no restrictions on starting new firms. Each firm can run
one factory.

a. Draw the supply curve for one firm; label it
Sf. Draw the
supply curve for the industry, label it Si. Da is the demand curve for
typewriters; everyone expects it to stay the same for ever. How many
are sold at what price? How many firms are there?

b.The demand curve shifts up to Db. It takes a year to build a
new typewriter factory. Draw the short run supply curve SSR, showing
price as a function of quantity over times too short to build more
factories. A month after the change, how many typewriters are sold at
what price?

c. AC on Figure 13-17a includes the cost of
building a typewriter factory, which is three million dollars.
Factories last for 10 years; the interest rate is zero and factories
have no scrap value. After the firms have adjusted to
Db, the word
processor is invented and the demand curve for typewriters suddently
shifts down to Dc. Everyone expects it to remain there forever. Immediately
after the change, what is the price of a typewriter?

d. The demand curve remains at Dc. Fifty years later, what is
the price of a typewriter? How many are produced each year?

4. Long-run total cost includes both short-run and
long-run expenses, so for any quantity long-run total cost must be
larger than short-run total cost. True or False? Discuss.

The following problems refer to the optional
section:

5. You have $30,000; your utility function is
shown by Figure 13-12. There is one chance in a hundred that your
house will be struck by lightning, in which case it will cost $10,000
to repair it. What is the highest price you would be willing to pay,
if necessary, for a lightning rod to protect your house?

6. Answer Problem 6 for the utility function of
Figure 13-13a.

7. Figure 13-18 is identical to Figure 13-13a,
with the addition of a line connecting two points--A and E--on the
utility function. I claim that point C, halfway between points A and
E, represents the utility (vertical axis) and expected return
(horizontal axis) of a fifty-fifty gamble between A ($10,000) and E
($30,000); the fact that C is below the graph of the utility function
indicates that you prefer a certainty with the same expected return
($20,000) to such a gamble. Similarly, I claim that point B
represents a gamble with a 75 percent chance of giving you A and a 25
percent chance of giving you E, and that point D represents a gamble
with a 25 percent chance of A and a 75 percent chance of E.

Prove that these claims are true--that the
vertical position of each point equals the expected utility of the
corresponding gamble and that the horizontal position equals the
expected return.

8. In the text, I asserted that declining marginal
utility of income was equivalent to risk aversion and that increasing
marginal utility of income was equivalent to risk preference. While I
gave examples, I did not prove that the assertion was true in
general. Use the result of Problem 10 to do so.

9. In discussing risk aversion, I have only
considered alternatives that are measured in money. Suppose you are
gambling in apples instead. Is it possible for someone to be a risk
preferrer in terms of dollars and a risk averter in terms of apples?
Vice versa? Does it depend on whether there is a market on which you
can buy and sell apples?

10. In one episode of Star Trek, Spock is in an
orbiting landing craft that is running out of fuel and will shortly
crash. Captain Kirk and the Enterprise are about to leave the planet,
having somehow misplaced one landing craft and science officer. Spock
fires his rockets, burning up all the remaining fuel, in the hope
that the Enterprise will notice the flare and come rescue him. Later
Kirk twits the supremely logical Spock with irrationality, for having
traded his last few hours of fuel for a one in a hundred chance of
rescue. Is Kirk correct? Was Spock's behavior irrational?

FOR FURTHER READING

The original discussion of Von Neumann utility is
in John Von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior (Princeton: Princeton University Press, 1944), Chapter
1.

For discussions of some of the philosophical
issues associated with what, if anything, the good society would
maximize, you may wish to look at two important books: Robert Nozick,
Anarchy, State and Utopia (New York: Basic Books, Inc., 1974) and John Rawls,
A Theory of Justice (Cambridge: Harvard University Press, 1971). You may also
be interested in an essay of mine on the question of what you should
maximize if one of the variables is the number of people; in that
situation maximizing total utility and maximizing average utility
lead to different results. It is:

"What Does Optimum Population Mean,"
Research in Population
Economics, Vol. III (1981), Eds. Simon and
Lindert.